The decomposition approach was refined by Chang and Saranurak [1]}, allowing the complexity of triangle listing to drop to \(\tilde{O}(n^{1/3})\)  rounds, w.h.p. Since the \(\tilde{\Omega }(n^{1/3})\)  lower bound for triangle listing in the \(\mathsf {CLIQUE}\)  model directly applies also in \(\mathsf {CONGEST}\) , this complexity is near-optimal (up to \(\operatorname{polylog}{n}\)  factors). At a high level, the decomposition obtained in [1]} improves upon the one in [3]} in that it does not have \(E_s\)  at all (the bounded arboricity part) and in the complexity of obtaining it. In addition, a somewhat modified version of the routing algorithms of [4]}, [5]} is introduced, for the triangle listing usage of the decomposition.
